VHIM

The method relies on characterizing the pipe and the subsonic valve by frictional losses in velocity heads. The valve frictional loss is given from equations (7.25) and (7.36) as 

Figur 10.6 Frictional loss in velocity heads (VHIM). 

VHIM calculates the valve throat to valve inlet pressure ratio to be greater than the critical value until 17 seconds into the transient, indicating that the 

It is noticeable that a small discontinuity is introduced between 17 and 18 seconds, caused by the 

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The remarks made in Section 8.1 translate across to the gas flow cases more or less word for word, except that the methods of Chapter 9 must now be allied to those of Chapter 6 in order to calculate gas flow through line and valve. But the more complicated equations for both line flow and valve flow render explicit solutions to the full set of equations impossible. Two implicit methods, the Velocity-Head Implicit Method (VHIM) and the Smoothed Velocity-Head Implicit Method (SVHIM), will be presented, where the solution process has been reduced to iteration on a single variable. The SVHIM is judged to be more accurate because it deals with the compressible-flow valve equations at all times. 

Gas flow through an installed valve - Velocity-Head Implicit Method (VHIM)... 

The methodology of Chapter 6 is applied in a fairly straightforward way in VHIM to estimate the flow as follows. First estimate the number of velocity heads, Kj, dropped over the upstream section of pipe using equation (6.58) from Section 6.6, repeated telow ... 

Detecting the onset of sonic flow in the valve using VHIM... 

Calculating the flow and pipe conditions when valve flow is sonic in the VHIM... 

The VHIM is able to deal with both choking at the end of the pipe and choking of the valve. Nevertheless,... 

The equations underlying ASVAM will now be set down for the plant arrangement of Figure 10.1. First we calculate the frictional loss in velocity heads in the same way as laid down for VHIM in Section 10.2 for the upstream section of pipe, the downstream section and the valve (equations (6.58), (7.36) and (7.25)), and then find the total frictional head loss, Kt, from equation (10.1). 

Figure 10.8 Pressures at various points in the pipe (VHIM).

The pressure transient calculated by VHIM is shown in Figure 10.8, which matches closely that plotted in Figure 10.4 for SVHIM. In particular, it will be seen that the pipe outlet pressure, p3, has reduced to atmospheric, p4, by 20 seconds, in agreement with SVHIM, indicating that the outlet flow is subsonic at this time. 

The resulting flow transient calculated by ASVAM is shown in Figure lO.lOcompared with that of VHIM. The flows are almost identical over the subsonic region up to time = 16 seconds, and come back together again aher 18 seconds. It is noticeable that the discontinuity between subsonic valve flow and sonic valve flow that characterizes VHIM disappears under ASVAM. This is because of the transition in ASVAM takes a very simple form, namely the maximum selection of equation (10.62). 

Figure 10.11 compares the flow transient calculated by ASVAM with the standard transient calculated by SVHIM. ASVAM, like VHIM, relies implicitly on the C value to characterize valve flow in the subsonic region via the calculation of AT, and then Kp. As a result, ASVAM underestimates the flow by about 3% at the beginning of the transient in the same way as VHIM. But ASVAM produces essentially the same value as SVHIM for flow by time = 15 seconds. In fact, ASVAM predicts sonic flow in the valve at time = 16 seconds, a second in advance of SVHIM, but the difference in flow is very small. 

VHIM is based on a simple transfer to the pipe-plus-valve case of the method outlined in Section 6.4 for calculating flow in a pipe. It will be less accurate than SVHIM in the subsonic-valve region because only the liquid valve coefficient, C , is used in this flow regime, rather than the more representative gas coefficient, Cg. This causes a small discontinuity to occur when sonic flow conditions are met in the valve, and the C characterization is superseded by a characterization based on Cg. The loss in accuracy compared with SVHIM is 3% or less, but VHIM retains the disadvantage that it requires an iterative solution. 

ASVAM is an essentially explicit approximation to VHIM. ASVAM benefits from the extensive offline computations carried out to define both the critical pressure ratio, P3c/Pi. as a function of the fric-tionsd loss in velocity heads, Kr, and also the shape of the bo versus Kt and Pa/pi surface. It is much quicker and less complicated than VHIM as a result. 

Note that the instrumental error in the construction of the individual kinetic curve in separate experiments is rather small we can observe this from the location of the points on the kinetic curve (see Figures 5.1-5.8). At the same time, comparison of the individual kinetic curves give the parameter scattering, calculated with the use of these kinetic curves (for instance, values Wo in Figures 5.19, 5.24, 5.25), that exceeds the error of individual experiments manifold. Hiis phenomenon is well-known for 3-D polymerization and is qualified as a bad reproduction of the kinetic measurements or vhims of the process. From our point of view, the above-mentioned characteristic of the 3-D polymerization is direct proof of the microheterogeneity of the process, its active role in it the Tiquid... 

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